
where \(\mathbf {0}_p\)  is \(p-\) vector with all entries being 0. Hence, \(R_n\)  can be seen as the remainder term in second order Taylor's expansion of \( k_{\mathcal {Z}}(\Delta _{\mathcal {Z}},{\bf X}_n)\)  around \(\Delta _{\mathcal {Z}}=\mathbf {0}_{p\times p}\) , viewing \( k_{\mathcal {Z}}(\Delta _{\mathcal {Z}},{\bf X}_n)\)  as a function of \(\tilde{\Delta }_{\mathcal {Z}}\)  with \({\bf X}_n\)  given. Then, following the argument in (A.14)-(A.16) of [1]}, we have
\(|R_n|&\le \frac{1}{2}||\Delta _{\mathcal {Z}}||_{\operatorname{F}}^2(p+\#\mathcal {Z})\max _{0\le u\le 1}||\operatorname{H}_{\Sigma ^*+u\Delta _{\mathcal {Z}}}-\operatorname{H}_{\Sigma _{\mathcal {Z}}^*}||_{\infty }.\) 
